take X ; :: thesis: ( X is reflexive & X is discerning & X is RealNormSpace-like & X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus X is reflexive by Lm5; :: thesis: ( X is discerning & X is RealNormSpace-like & X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus X is discerning by Lm5; :: thesis: ( X is RealNormSpace-like & X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus X is RealNormSpace-like by Lm5; :: thesis: ( X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital & X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
thus ( X is vector-distributive & X is scalar-distributive & X is scalar-associative & X is scalar-unital ) :: thesis: ( X is Abelian & X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
proof
thus for a being Real
for v, w being VECTOR of X holds a * (v + w) = (a * v) + (a * w) :: according to RLVECT_1:def 5 :: thesis: ( X is scalar-distributive & X is scalar-associative & X is scalar-unital )
proof
let a be Real; :: thesis: for v, w being VECTOR of X holds a * (v + w) = (a * v) + (a * w)
let v, w be VECTOR of X; :: thesis: a * (v + w) = (a * v) + (a * w)
reconsider v9 = v, w9 = w as VECTOR of ((0). the RealLinearSpace) ;
thus a * (v + w) = a * (v9 + w9)
.= (a * v9) + (a * w9) by RLVECT_1:def 5
.= (a * v) + (a * w) ; :: thesis: verum
end;
thus for a, b being Real
for v being VECTOR of X holds (a + b) * v = (a * v) + (b * v) :: according to RLVECT_1:def 6 :: thesis: ( X is scalar-associative & X is scalar-unital )
proof
let a, b be Real; :: thesis: for v being VECTOR of X holds (a + b) * v = (a * v) + (b * v)
let v be VECTOR of X; :: thesis: (a + b) * v = (a * v) + (b * v)
reconsider v9 = v as VECTOR of ((0). the RealLinearSpace) ;
thus (a + b) * v = (a + b) * v9
.= (a * v9) + (b * v9) by RLVECT_1:def 6
.= (a * v) + (b * v) ; :: thesis: verum
end;
thus for a, b being Real
for v being VECTOR of X holds (a * b) * v = a * (b * v) :: according to RLVECT_1:def 7 :: thesis: X is scalar-unital
proof
let a, b be Real; :: thesis: for v being VECTOR of X holds (a * b) * v = a * (b * v)
let v be VECTOR of X; :: thesis: (a * b) * v = a * (b * v)
reconsider v9 = v as VECTOR of ((0). the RealLinearSpace) ;
thus (a * b) * v = (a * b) * v9
.= a * (b * v9) by RLVECT_1:def 7
.= a * (b * v) ; :: thesis: verum
end;
let v be VECTOR of X; :: according to RLVECT_1:def 8 :: thesis: 1 * v = v
reconsider v9 = v as VECTOR of ((0). the RealLinearSpace) ;
thus 1 * v = 1 * v9
.= v by RLVECT_1:def 8 ; :: thesis: verum
end;
A1: for x, y being VECTOR of X
for x9, y9 being VECTOR of ((0). the RealLinearSpace) st x = x9 & y = y9 holds
( x + y = x9 + y9 & ( for a being Real holds a * x = a * x9 ) ) ;
thus for v, w being VECTOR of X holds v + w = w + v :: according to RLVECT_1:def 2 :: thesis: ( X is add-associative & X is right_zeroed & X is right_complementable & X is strict )
proof
let v, w be VECTOR of X; :: thesis: v + w = w + v
reconsider v9 = v, w9 = w as VECTOR of ((0). the RealLinearSpace) ;
thus v + w = w9 + v9 by A1
.= w + v ; :: thesis: verum
end;
thus for u, v, w being VECTOR of X holds (u + v) + w = u + (v + w) :: according to RLVECT_1:def 3 :: thesis: ( X is right_zeroed & X is right_complementable & X is strict )
proof
let u, v, w be VECTOR of X; :: thesis: (u + v) + w = u + (v + w)
reconsider u9 = u, v9 = v, w9 = w as VECTOR of ((0). the RealLinearSpace) ;
thus (u + v) + w = (u9 + v9) + w9
.= u9 + (v9 + w9) by RLVECT_1:def 3
.= u + (v + w) ; :: thesis: verum
end;
thus for v being VECTOR of X holds v + (0. X) = v :: according to RLVECT_1:def 4 :: thesis: ( X is right_complementable & X is strict )
proof
let v be VECTOR of X; :: thesis: v + (0. X) = v
reconsider v9 = v as VECTOR of ((0). the RealLinearSpace) ;
thus v + (0. X) = v9 + (0. ((0). the RealLinearSpace))
.= v ; :: thesis: verum
end;
thus X is right_complementable :: thesis: X is strict
proof
let v be VECTOR of X; :: according to ALGSTR_0:def 16 :: thesis: v is right_complementable
reconsider v9 = v as VECTOR of ((0). the RealLinearSpace) ;
consider w9 being VECTOR of ((0). the RealLinearSpace) such that
A2: v9 + w9 = 0. ((0). the RealLinearSpace) by ALGSTR_0:def 11;
reconsider w = w9 as VECTOR of X ;
take w ; :: according to ALGSTR_0:def 11 :: thesis: v + w = 0. X
thus v + w = 0. X by A2; :: thesis: verum
end;
thus X is strict ; :: thesis: verum